List of top Questions asked in KEAM- 2025

If $e^{y}+x^{2}y+xy^{2}=e^{1}$, then $\frac{dy}{dx}$ at (0,1) is equal to ________.

The function $f(x)=|x^{2}-3x+2|,x\in\mathbb{R}$ is not differentiable at ________.

Let \([a]\) denote the greatest integer less than or equal to \(a\). Then \[ \lim_{x\to 0^{+}} x\left( \left[\frac{1}{x}\right] + \left[\frac{2}{x}\right] \right) \] is equal to ________.

$\lim_{x\rightarrow0}\frac{x \cos^{2}x}{\sin x}$ is equal to ________.

$\lim_{x\rightarrow2}\frac{(x^{3}-8)\sin(x-2)}{x^{2}-4x+4}$ is equal to ________.

$\lim_{x\rightarrow0}\frac{\sin(\pi \sin^{2}x)}{x^{2}} = $ ________.

Four unbiased coins are tossed simultaneously. Probability of getting at most two heads is ________.

The variance of 240, 260, 270, 280 is ________.

If $P(A)=0.7, P(B)=0.5$ and $P(A\cup B)=0.9$, then $P(A/B)$ is ________.

A straight line through the point (1,-1,0) meets the line $\frac{x-1}{1}=\frac{y+1}{1}=\frac{z-1}{-1}$ at right angle. Its equation is ________.

Which one of the following is a vector parallel to the straight line $\vec{r}=(\hat{i}-11\hat{j}+101\hat{k})+\lambda(3\hat{i}-5\hat{j}+2\hat{k}),\lambda\in\mathbb{R}$? ________.

The equation of the line passing through (0, 0, 1) and (1, 1, 0) is ________.

The point of intersection of the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-11}{4}$ and $\frac{x-3}{1}=\frac{y-\frac{9}{2}}{2}=\frac{z}{1}$ is ________.

Let $\vec{a}\times(2\hat{i}+3\hat{j}+4\hat{k})=(2\hat{i}+3\hat{j}+4\hat{k})\times\vec{b}$. If $|\vec{a}+\vec{b}|=\sqrt{29}$, then $\vec{a}+\vec{b} = $ ________.

The line $x-y+4=0$ touches the ellipse $x^{2}+3y^{2}=12$ at ________.

The centre of the ellipse $4x^{2}+24x+9y^{2}-18y+9=0$ is ________.

The length of the latus rectum of the ellipse \[ \frac{x^2}{9}+\frac{y^2}{16}=1 \] is ________.

The axis of a parabola is $x=0$. If the vertex is at a distance 3 from the origin above the x-axis, the vertex of the parabola is at ________.

Which one of the following lines passes through the point of intersection of $x+y=5$ and $2x+y=7$? ________.

Let $P(1,2)$, $Q(a,b)$, $R(5,7)$ and $S(2,3)$ be the vertices of a parallelogram $PQRS$. Then ________.

Let $a \ne 1$ be a non-zero real number. If the lines $2x+ay=1$ and $x+2y=1$ are perpendicular, then the value of $a$ is equal to ________.

$\tan^{-1}(\frac{1001}{999}) - \tan^{-1}(\frac{2}{2000}) = $ ________.

If $\sec^{-1}\left(\frac{x}{x+2}\right) = \frac{\pi}{2} - \csc^{-1}\left(\frac{1}{2}\right)$, then $x = $ ________.

$\sec(\cos^{-1}(\frac{2024}{2025}))$ is equal to ________.

$\frac{\sin\frac{\pi}{7}+\sin\frac{2\pi}{7}}{1+\cos\frac{\pi}{7}+\cos\frac{2\pi}{7}} = $ ________.